Solving for x in the Equation (2^x cdot frac{1}{x} 4): A Comprehensive Guide

Imagine you are presented with the equation (2^x cdot frac{1}{x} 4) and are tasked with finding the value of (x). This article delves into the various methods of solving this problem, providing a clear and detailed explanation for each step. By the end of this guide, you will have a thorough understanding of the algebraic manipulation, logarithmic laws, and quadratic equations involved.

Introduction

This equation is an interesting problem that can be solved through a mix of algebraic manipulation and logical reasoning. The goal is to isolate (x) and determine its value. We will explore multiple methods to find the solution, each providing unique insights into mathematical problem-solving.

Solving via Direct Substitution

One straightforward method to solve this equation is by direct substitution. By adding the definition of (A 2^x - frac{1}{x}) and (B 2^{frac{1}{x}} cdot x), assuming (x frac{1}{x}) (which simplifies to (x 1)), we can verify that this value satisfies the equation. This method relies on basic algebraic checks and confirms the value of (x).

Algebraic Verification

Direct Verification

When (x 1), we substitute this value into the equation:

[left(2^1 cdot frac{1}{1}right) cdot left(2^{frac{1}{1}} cdot 1right) 2 cdot 2 4]

This confirms that (x 1) is indeed a solution to the equation. This verification can be done directly, and the outcome aligns perfectly with the right-hand side (RHS) of the equation.

Using Inequality Properties

We can also use inequality properties to confirm the solution. The inequality (a^2b^2 geq 2ab) holds, and equality occurs if and only if (a b). Applying this to our equation:

[left(2^x cdot frac{1}{x}right) cdot left(2^{frac{1}{x}} cdot xright) geq 2 sqrt{left(2^x cdot frac{1}{x}right) cdot left(2^{frac{1}{x}} cdot xright)} 2 sqrt{2^x cdot 2^{frac{1}{x}}} 2^{1 frac{1}{x} / 2}]

We need to check when the equality holds:

[left(2^x cdot frac{1}{x}right) left(2^{frac{1}{x}} cdot xright)]

This implies:

[x frac{1}{x}]

Solving (x frac{1}{x}), we find (x 1).

Therefore, the only solution to the equation is (x 1).

Additional Insights

The problem can also be approached using logarithmic laws and quadratic equations. Taking the logarithm on both sides of the original equation, we can simplify using logarithmic properties. However, the specific steps in this approach may require more detailed algebraic manipulation.

One key finding is that the quadratic equation resulting from the logarithmic transformation can be easily solved by the quadratic formula. This method, while more complex, provides a deeper understanding of the underlying mathematical principles.

Conclusion

In conclusion, the value of (x) in the equation (2^x cdot frac{1}{x} 4) is (1). This solution can be verified through multiple methods, including direct substitution, algebraic verification, and logarithmic transformations. Each method offers unique insights and can be helpful in different contexts. Whether you prefer a quick verification or a deeper understanding of the problem, this guide has covered all the necessary steps.